For problems of pair-production in a given field, it has been common to neglect the perpendicular momentum dependence, see e.g. Refs. [1]}. While this is a rather natural simplification, as the perpendicular momentum merely adds some extra mass to the pairs, nevertheless the accuracy of this approximation might not be very high. Studying Eqs. (REF ) for a given electric pulse with a temporal sech-profile, it is found that the approximation often is a useful one. Nevertheless, it is somewhat problematic to omit the perpendicular momentum dependence, as the error in the pair-production rate induced by this omission depends on the parameters of the problem. Specifically, for weakly inhomogeneous systems (such that the local density approximation is applicable), the perpendicular momentum of the generated pairs is close to linearly proportional to the electric field (cf. fig 3.) As a result, there is a general overestimation of the produced pairs when the perpendicular momentum is overlooked. While, in principle, a correction factor could be introduced to compensate for the overestimation, such a solution is not entirely satisfactory, as the correction factor would be dependent on the electric field magnitude, that could be varying dynamically in a self-consistent field model.
